canonical form of element of number field
Theorem. Let ϑ be an algebraic number
of degree (http://planetmath.org/DegreeOfAnAlgebraicNumber) n. Any element α of the algebraic number field
ℚ(ϑ) may be uniquely expressed in the canonical form
α=c0+c1ϑ+c2ϑ2+…+cn-1ϑn-1 | (1) |
where the numbers ci are rational.
Proof. We start from the fact that ℚ(ϑ) consists of all expressions formed of ϑ and rational numbers using arithmetic operations (no divisor (http://planetmath.org/Division) must vanish); such expressions lead always to the form
α=a(ϑ)b(ϑ) | (2) |
where the numerator and the denominator are polynomials in ϑ with rational coefficients (which can, in fact, be chosen integers).
So, let α in (2) an arbitrary element of the field ℚ(ϑ). Denote by f(x) the minimal polynomial of ϑ over ℚ. Since b(ϑ)≠0, the polynomial f(x) does not divide (http://planetmath.org/DivisibilityInRings) b(x), and since f(x) is irreducible (http://planetmath.org/IrreduciblePolynomial2), the greatest common divisor
(http://planetmath.org/PolynomialRingOverFieldIsEuclideanDomain) of f(x) and b(x) is a constant polynomial, which can be normed to 1. Thus there exist the polynomials φ(x) and ψ(x) of the ring ℚ[x] such that
φ(x)f(x)+ψ(x)b(x)≡ 1. |
Especially
φ(ϑ)f(ϑ)⏟= 0+ψ(ϑ)b(ϑ)= 1, |
whence
1b(ϑ)=ψ(ϑ) |
and consequently
α=a(ϑ)b(ϑ)=a(ϑ)ψ(ϑ):= |
Hence, is a polynomial in with rational coefficients.
Let now
Denote
It follows that
whence (1) is true.
Suppose that we had also
with every rational. This implies that
i.e. that satisfies the equation
with rational coefficients and degree less than . Because the degree of is , it is possible only if all differences vanish. Thus
i.e. the (1) is unique.
Note 1. The polynomial is called the canonical polynomial of the algebraic number with respect to the primitive element (http://planetmath.org/SimpleFieldExtension) .
Note 2. The theorem allows to denote the field similarly as polynomial rings:
.
Note 3. When allowed, unlike in (1), higher powers of the primitive element (whose minimal polynomial is ), one may unlimitedly write different sum of , e.g.
Title | canonical form of element of number field |
Canonical name | CanonicalFormOfElementOfNumberField |
Date of creation | 2013-03-22 19:08:00 |
Last modified on | 2013-03-22 19:08:00 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 17 |
Author | pahio (2872) |
Entry type | Theorem |
Classification | msc 11R04 |
Related topic | Canonical |
Related topic | SimpleFieldExtension |
Related topic | CanonicalBasis |
Related topic | IntegralBasis |
Defines | canonical form |
Defines | canonical form in number field |
Defines | canonical polynomial |